On radial behaviour and balanced Bloch functions

Abstract

A Bloch function $g$ is a function analytic in the unit disk such that $(1–|z{|}^2)|g^{\prime }(z)|$ is bounded. First we generalize the theorem of Rohde that, for every "bad" Bloch function, $g(r\zeta )(r⟶1)$ follows any prescribed curve at a bounded distance for $\zeta$ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that $|g^{\prime }(z)|$ does not vary much on each circle $\{|z|=r\}$ except for small exceptional arcs. We show e.g. that

holds either for all $\zeta \in \mathbb{T}$ or for none.